When you heat up a material, it may change state. The molecules vibrate with a greater amplitude, and break apart from one another. The material has been supplied with energy and you can feel it getting hotter. The increased kinetic and potential (from their greater separation) energy of the particles is an increase in what we call internal energy. Internal energy is defined as:

The internal energy of a system is the sum of the randomly distributed kinetic and potential energies of its molecules.

Therefore, an increase in temperature for a material means an increase in its internal energy.

The Celsius scale of temperature depends on the properties of water. 0°C is the freezing point of water, and 100°C is the boiling point of water. It is a relative scale, because it is relative to the freezing and boiling points of water. The thermodynamic scale of temperature (represented by the letter T), however, is an absolute scale of temperuture, and does not depend on the properties of any particular substance. It is also directly proportional to the amount of internal energy a substance possesses.

This scale of temperature is defined in terms of internal energy, and is measured in kelvins (K). 0K is defined as the temperature at which a substance will have minimum internal energy, and is the lowest possible temperature. This temperature is known as absolute zero. It is at this point at which molecule stop vibrating and electrons stop spinning and orbiting.

If we were to heat a block of ice at a steady rate and plot a graph of the temperature against time, we would get the following graph:

This shape is rather surprising. You would expect the line to increase in a straight line, with none of the breaks that you can see above. We should consider what is happening to the molecules of the water at each section of the graph to understand why this is so:

AB

The ice is below freezing point, but the temperature is increasing. The molecules are vibrating slowly, but begin to vibrate more.

BC

At 273K (0°C) the ice is at melting point. The bonds between molecules are being broken and molecules have greater potential energy. This is the Latent Heat of Fusion

CD

The water now increases in temperature towards boiling point. The molecules vibrate even more and move around rapidly as their kinetic energy increases.

DE

At 373K (100°C) the water is now at boiling point. Molecules completely break away from each other and their potential energy increases. DE is much larger than BC because ALL bonds need to be broken for a gas to form. (The Latent Heat of Vapourisation.)

EF

The water is now steam and the molecules are moving around much faster than before. Their kinetic energy continues to increase as energy is supplied.

At the sections BC and DE, where there is a change of state, the molcules do not increase in kinetic energy, but increase in potential energy. The heat energy being supplied does not change the temperature at these sections, but is instead used to break the bonds between molecules.

Some materials will heat up quicker than others. For example, metals are good conductors of heat, and provided they are the same mass and that the energy is supplied at the same rate, copper will increase in temperature quicker than water.

The specific heat capacity can tell us how much energy is required to increase the temperature of a substance, and is defined as:

The specific heat capacity of a substance is numerically equal to the amount of energy required to raise the temperature of 1kg of the substance by 1K (or by 1°C).

This can be represented as:

c=QmΔt{\displaystyle c={\frac {Q}{m\Delta t}}}

Or rewritten,

Q=mcΔT{\displaystyle {Q}=mc\Delta T}

Where Q{\displaystyle {Q}} is the energy supplied, m{\displaystyle m} is the mass of the substance, c{\displaystyle c} is the specific heat capacity, and ΔT{\displaystyle \Delta T} is the change in temperature

To find the specific heat capacity of something, we can control all of the possible variables and then use them to calculate it. From the equation above, we can see that c=QmΔT{\displaystyle c={\frac {Q}{m\Delta T}}}. This means that if we can supply a known amount of energy to a material of known mass, and measure the change in temperature, we can insert the values into the equation and obtain the specific heat capacity.

To supply a known amount of energy, we can use an electric heater. You may recall that electrical energy can be found by E=IVt{\displaystyle E=IVt}, so by measuring the voltage, the current and the time that the circuit is switched on, we will have a value for the energy supplied to the material.

In the same time period that the circuit is switched on, we must take measurments for the change in temperature. An ordinary mercury thermometer may be used, although it is recommend to use a temperature sensor with a computer to make more precise and accurate measurements.

Once we have taken readings of the temperature and energy at regular intervals of time, we can plot a graph of Q{\displaystyle Q} against ΔT{\displaystyle \Delta T}. We can calculate the gradient, making sure to use as much of the line in our calculation as possible, and divide it by the mass of the material to obtain the value of the materials specific heat capacity.

When you heat up a substance so that it changes state, the temperature stays the same during the change. Different substances will require more energy to change state than others. The specific latent heat will tell us how much energy a substance requires to change state and is defined as:

The specific latent heat of a substance is numerically equal to the energy that must be supplied to change the state of 1kg of the substance without any change in temperature.

This can be written as the equation:

l=Qm{\displaystyle {l}={\frac {Q}{m}}}

Or, rewritten,

Q=ml{\displaystyle Q=ml}

Where Q{\displaystyle Q} is the energy supplied, m{\displaystyle m} is the mass of the substance, and L{\displaystyle L} is the specific latent heat.

There are four properties of a gas, that are related to each other. These properties are the pressure, the temperature, the volume and the mass of the gas, and these relationships are expressed as the gas laws.

Boyle's law relates the pressure of a gas to its volume. Specifically, it states that:

The pressure of a fixed mass of gas is inversely proportional to its volume, provided that the temperature remains constant.

This can be expressed as p∝1V{\displaystyle p\propto {\frac {1}{V}}} or pV=constant{\displaystyle pV=constant}.

You can picture this at the molecular level, if you were to imagine the number of collisions the particles of a gas make with the container of a particular size, and then imagine the increased number of collisions when the container is reduced in size but the number of particles remain the same. This is observed as an increase in pressure of the gas.

Charles' law relates the volume of a gas with its temperature on the thermodynamic temperature scale, and that:

The volume of a fixed mass of gas at constant pressure is proportional to its temperature on the thermodynamic temperature scale.

This can be expressed as V∝T{\displaystyle V\propto T} or VT=constant{\displaystyle {\frac {V}{T}}=constant}.

It is a little more difficult to understand why this is the case, because a gas will always take up the entire volume of its container. If you think about how a particle behaves when it is heated up, it will vibrate more and cause an increase in pressure, or harder and faster collisions of the molecules against the container. However, since pressure is to be kept constant in this case, the volume of the container will need to increase. Therefore by increasing the temperature of the gas, we have increased its volume.